## Involve: A Journal of Mathematics

• Involve
• Volume 11, Number 3 (2018), 489-500.

### A tale of two circles: geometry of a class of quartic polynomials

#### Abstract

Let $P$ be the family of complex-valued polynomials of the form $p(z)=(z−1)(z−r1)(z−r2)2$ with $|r1|=|r2|=1$. The Gauss–Lucas theorem guarantees that the critical points of $p∈P$ will lie within the unit disk. This paper further explores the location and structure of these critical points. For example, the unit disk contains two “desert” regions, the open disk ${z∈ℂ:|z−34|<14}$ and the interior of $2x4−3x3+x+4x2y2−3xy2+2y4=0$, in which critical points of $p$ cannot occur. Furthermore, each $c$ inside the unit disk and outside of the two desert regions is the critical point of at most two polynomials in $P$.

#### Article information

Source
Involve, Volume 11, Number 3 (2018), 489-500.

Dates
Received: 21 February 2017
Revised: 5 June 2017
Accepted: 13 June 2017
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.involve/1513775079

Digital Object Identifier
doi:10.2140/involve.2018.11.489

Mathematical Reviews number (MathSciNet)
MR3733970

Zentralblatt MATH identifier
06817033

#### Citation

Frayer, Christopher; Gauthier, Landon. A tale of two circles: geometry of a class of quartic polynomials. Involve 11 (2018), no. 3, 489--500. doi:10.2140/involve.2018.11.489. https://projecteuclid.org/euclid.involve/1513775079

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